|Résumé||A successful idea in the area of verification is to consider finite-state abstractions of infinite-state systems. A prominent example is the fact that many language classes satisfy a Parikh's theorem, i.e. for each language, there exists a finite automaton that accepts the same language up to the order of letters. Hence, provided that the abstraction preserves pertinent properties, this allows us to work with finite-state systems, which are much easier to handle.
While Parikh-style abstractions have been studied very intensely over the last decades, recent years have seen an increasing interest in abstractions based on the subword ordering. Examples include the set of (non necessarily contiguous) subwords of members of a language (the downward closure), or their superwords (the upward closure). Whereas it is well-known that these closures are regular for any language, it is often not obvious how to compute them. Another type of subword based abstractions are piecewise testable separators. Here, a separators acts as an abstraction of a pair of languages.
This talk will present approaches to computing closures, deciding separability by piecewise testable languages, and a (perhaps surprising) connection between these problems. If time permits, complexity issues will be discussed as well. |